Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit disc

TL;DR

This paper proves that the tangential speed of parabolic semigroups of holomorphic self-maps in the unit disc is asymptotically bounded by (1/2)logt, confirming a conjecture by Bracci.

Contribution

It establishes an asymptotic upper bound for the tangential speed of parabolic semigroups, advancing understanding of their boundary behavior.

Findings

Tangential speed is asymptotically bounded by (1/2)logt.

Proves a conjecture by Bracci on speed bounds.

Introduces a result on asymptotic monotonicity for proper pairs of semigroups.

Abstract

We show that the tangential speed of a parabolic semigroup of holomorphic self-maps in the unit disc is asymptotically bounded from above by (1/2)logt, proving a conjecture by Bracci. In order to show the proof we need a result of "asymptotical monotonicity" of the tangential speed for proper pairs of parabolic semigroups with positive hyperbolic step.